A heat kernel approach to the stability of exponential equation in distributions and hyperfunctions

Abstract

Let f,g,h:G×(0,∞)→C with G an Abelian group. We consider the stability of the Pexiderized exponential equation, |f(x+y,t+s)−g(x,t)h(y,s)|≤H(t,s),x,y∊G, t,s>0, where H:(0,∞)2→(0,∞) is a decreasing function. As an application we establish the stability problem of the Pexiderized exponential equation, u∘A−v⊗w∊D′L∞(R2n), in some spaces of generalized functions, where D′L∞(R2n) denotes the space of bounded distributions.